20.19 Cohomology and colimits

Let $X$ be a ringed space. Let $(\mathcal{F}_ i, \varphi _{ii'})$ be a system of sheaves of $\mathcal{O}_ X$-modules over the directed set $I$, see Categories, Section 4.21. Since for each $i$ there is a canonical map $\mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ we get a canonical map

$\mathop{\mathrm{colim}}\nolimits _ i H^ p(X, \mathcal{F}_ i) \longrightarrow H^ p(X, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i)$

for every $p \geq 0$. Of course there is a similar map for every open $U \subset X$. These maps are in general not isomorphisms, even for $p = 0$. In this section we generalize the results of Sheaves, Lemma 6.29.1. See also Modules, Lemma 17.22.8 (in the special case $\mathcal{G} = \mathcal{O}_ X$).

Lemma 20.19.1. Let $X$ be a ringed space. Assume that the underlying topological space of $X$ has the following properties:

1. there exists a basis of quasi-compact open subsets, and

2. the intersection of any two quasi-compact opens is quasi-compact.

Then for any directed system $(\mathcal{F}_ i, \varphi _{ii'})$ of sheaves of $\mathcal{O}_ X$-modules and for any quasi-compact open $U \subset X$ the canonical map

$\mathop{\mathrm{colim}}\nolimits _ i H^ q(U, \mathcal{F}_ i) \longrightarrow H^ q(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i)$

is an isomorphism for every $q \geq 0$.

Proof. It is important in this proof to argue for all quasi-compact opens $U \subset X$ at the same time. The result is true for $q = 0$ and any quasi-compact open $U \subset X$ by Sheaves, Lemma 6.29.1 (combined with Topology, Lemma 5.27.1). Assume that we have proved the result for all $q \leq q_0$ and let us prove the result for $q = q_0 + 1$.

By our conventions on directed systems the index set $I$ is directed, and any system of $\mathcal{O}_ X$-modules $(\mathcal{F}_ i, \varphi _{ii'})$ over $I$ is directed. By Injectives, Lemma 19.5.1 the category of $\mathcal{O}_ X$-modules has functorial injective embeddings. Thus for any system $(\mathcal{F}_ i, \varphi _{ii'})$ there exists a system $(\mathcal{I}_ i, \varphi _{ii'})$ with each $\mathcal{I}_ i$ an injective $\mathcal{O}_ X$-module and a morphism of systems given by injective $\mathcal{O}_ X$-module maps $\mathcal{F}_ i \to \mathcal{I}_ i$. Denote $\mathcal{Q}_ i$ the cokernel so that we have short exact sequences

$0 \to \mathcal{F}_ i \to \mathcal{I}_ i \to \mathcal{Q}_ i \to 0.$

We claim that the sequence

$0 \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i \to \mathop{\mathrm{colim}}\nolimits _ i \mathcal{Q}_ i \to 0.$

is also a short exact sequence of $\mathcal{O}_ X$-modules. We may check this on stalks. By Sheaves, Sections 6.28 and 6.29 taking stalks commutes with colimits. Since a directed colimit of short exact sequences of abelian groups is short exact (see Algebra, Lemma 10.8.8) we deduce the result. We claim that $H^ q(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = 0$ for all quasi-compact open $U \subset X$ and all $q \geq 1$. Accepting this claim for the moment consider the diagram

$\xymatrix{ \mathop{\mathrm{colim}}\nolimits _ i H^{q_0}(U, \mathcal{I}_ i) \ar[d] \ar[r] & \mathop{\mathrm{colim}}\nolimits _ i H^{q_0}(U, \mathcal{Q}_ i) \ar[d] \ar[r] & \mathop{\mathrm{colim}}\nolimits _ i H^{q_0 + 1}(U, \mathcal{F}_ i) \ar[d] \ar[r] & 0 \ar[d] \\ H^{q_0}(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) \ar[r] & H^{q_0}(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{Q}_ i) \ar[r] & H^{q_0 + 1}(U, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i) \ar[r] & 0 }$

The zero at the lower right corner comes from the claim and the zero at the upper right corner comes from the fact that the sheaves $\mathcal{I}_ i$ are injective. The top row is exact by an application of Algebra, Lemma 10.8.8. Hence by the snake lemma we deduce the result for $q = q_0 + 1$.

It remains to show that the claim is true. We will use Lemma 20.11.9. Let $\mathcal{B}$ be the collection of all quasi-compact open subsets of $X$. This is a basis for the topology on $X$ by assumption. Let $\text{Cov}$ be the collection of finite open coverings $\mathcal{U} : U = \bigcup _{j = 1, \ldots , m} U_ j$ with each of $U$, $U_ j$ quasi-compact open in $X$. By the result for $q = 0$ we see that for $\mathcal{U} \in \text{Cov}$ we have

$\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = \mathop{\mathrm{colim}}\nolimits _ i \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}_ i)$

because all the multiple intersections $U_{j_0 \ldots j_ p}$ are quasi-compact. By Lemma 20.11.1 each of the complexes in the colimit of Čech complexes is acyclic in degree $\geq 1$. Hence by Algebra, Lemma 10.8.8 we see that also the Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i)$ is acyclic in degrees $\geq 1$. In other words we see that $\check{H}^ p(\mathcal{U}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{I}_ i) = 0$ for all $p \geq 1$. Thus the assumptions of Lemma 20.11.9 are satisfied and the claim follows. $\square$

Next we formulate the analogy of Sheaves, Lemma 6.29.4 for cohomology. Let $X$ be a spectral space which is written as a cofiltered limit of spectral spaces $X_ i$ for a diagram with spectral transition morphisms as in Topology, Lemma 5.24.5. Assume given

1. an abelian sheaf $\mathcal{F}_ i$ on $X_ i$ for all $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$,

2. for $a : j \to i$ an $f_ a$-map $\varphi _ a : \mathcal{F}_ i \to \mathcal{F}_ j$ of abelian sheaves (see Sheaves, Definition 6.21.7)

such that $\varphi _ c = \varphi _ b \circ \varphi _ a$ whenever $c = a \circ b$. Set $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits p_ i^{-1}\mathcal{F}_ i$ on $X$.

Lemma 20.19.2. In the situation discussed above. Let $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and let $U_ i \subset X_ i$ be quasi-compact open. Then

$\mathop{\mathrm{colim}}\nolimits _{a : j \to i} H^ p(f_ a^{-1}(U_ i), \mathcal{F}_ j) = H^ p(p_ i^{-1}(U_ i), \mathcal{F})$

for all $p \geq 0$. In particular we have $H^ p(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(X_ i, \mathcal{F}_ i)$.

Proof. The case $p = 0$ is Sheaves, Lemma 6.29.4.

In this paragraph we show that we can find a map of systems $(\gamma _ i) : (\mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ with $\mathcal{G}_ i$ an injective abelian sheaf and $\gamma _ i$ injective. For each $i$ we pick an injection $\mathcal{F}_ i \to \mathcal{I}_ i$ where $\mathcal{I}_ i$ is an injective abelian sheaf on $X_ i$. Then we can consider the family of maps

$\gamma _ i : \mathcal{F}_ i \longrightarrow \prod \nolimits _{b : k \to i} f_{b, *}\mathcal{I}_ k = \mathcal{G}_ i$

where the component maps are the maps adjoint to the maps $f_ b^{-1}\mathcal{F}_ i \to \mathcal{F}_ k \to \mathcal{I}_ k$. For $a : j \to i$ in $\mathcal{I}$ there is a canonical map

$\psi _ a : f_ a^{-1}\mathcal{G}_ i \to \mathcal{G}_ j$

whose components are the canonical maps $f_ b^{-1}f_{a \circ b, *}\mathcal{I}_ k \to f_{b, *}\mathcal{I}_ k$ for $b : k \to j$. Thus we find an injection $\{ \gamma _ i\} : \{ \mathcal{F}_ i, \varphi _ a) \to (\mathcal{G}_ i, \psi _ a)$ of systems of abelian sheaves. Note that $\mathcal{G}_ i$ is an injective sheaf of abelian groups on $X_ i$, see Lemma 20.11.11 and Homology, Lemma 12.27.3. This finishes the construction.

Arguing exactly as in the proof of Lemma 20.19.1 we see that it suffices to prove that $H^ p(X, \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{G}_ i) = 0$ for $p > 0$.

Set $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{G}_ i$. To show vanishing of cohomology of $\mathcal{G}$ on every quasi-compact open of $X$, it suffices to show that the Čech cohomology of $\mathcal{G}$ for any covering $\mathcal{U}$ of a quasi-compact open of $X$ by finitely many quasi-compact opens is zero, see Lemma 20.11.9. Such a covering is the inverse by $p_ i$ of such a covering $\mathcal{U}_ i$ on the space $X_ i$ for some $i$ by Topology, Lemma 5.24.6. We have

$\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{G}) = \mathop{\mathrm{colim}}\nolimits _{a : j \to i} \check{\mathcal{C}}^\bullet (f_ a^{-1}(\mathcal{U}_ i), \mathcal{G}_ j)$

by the case $p = 0$. The right hand side is a filtered colimit of complexes each of which is acyclic in positive degrees by Lemma 20.11.1. Thus we conclude by Algebra, Lemma 10.8.8. $\square$

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